Counting compositions over finite abelian groups

TL;DR

This paper develops methods to count compositions over finite abelian groups with specific restrictions, connecting classical problems like Waring's problem and Carlitz compositions through algebraic and combinatorial techniques.

Contribution

It introduces a unified approach using multisection, transfer matrix, and Perron-Frobenius methods to count restricted compositions over finite abelian groups, linking various classical problems.

Findings

Exact formulas for counting restricted compositions

Asymptotic formulas derived using Perron-Frobenius theorem

Bijections between classes of restricted compositions

Abstract

We find the number of compositions over finite abelian groups under two types of restrictions: (i) each part belongs to a given subset and (ii) small runs of consecutive parts must have given properties. Waring's problem over finite fields can be converted to type~(i) compositions, whereas Carlitz and locally Mullen compositions can be formulated as type~(ii) compositions. We use the multisection formula to translate the problem from integers to group elements, the transfer matrix method to do exact counting, and finally the Perron-Frobenius theorem to derive asymptotics. We also exhibit bijections involving certain restricted classes of compositions.